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Mathematics 2 for Business Schools

Section 1: Fundamentals of Differential Calculus

Spring Semester 2017

After finishing this section you should be able to …

• derive the difference quotient and the differential quotient of a function (repetition).

• explain the concept of the derivative of functions (repetition).

• derive and correctly apply the rule for the derivative of constant functions (repetition).

• derive and correctly apply the rule for the derivative of power functions (repetition).

• correctly apply the constant factor rule and the sum rule (repetition).

• correctly apply the product rule, the quotient rule, and the chain rule (new).

• find the derivative of exponential functions and logarithmic functions (new).

Learning objectives

2Spring semester 2017 Section 1: Fundamentals of differential calculus

Difference quotient – Definition

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The slope 𝑚𝑠 of the secant through the

points

𝑃 𝑥0, 𝑓(𝑥0) and

𝑄 𝑥0 + Δ𝑥, 𝑓(𝑥0 + Δ𝑥)

of 𝑓, i.e., the average rate of change

of 𝑓 on the interval 𝑥0; 𝑥0 + Δ𝑥 is

called the difference quotient

𝑚𝑠 =Δ𝑓

Δ𝑥=𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0

Δ𝑥

at 𝑥0 (between 𝑃 and 𝑄).

Spring semester 2017 Section 1: Fundamentals of differential calculus

𝑥0 𝑥0 + ∆𝑥

𝑓(𝑥0)

𝑓(𝑥0 + ∆𝑥)

𝑃

𝑄

Δ𝑓

Δ𝑥

Local rate of change of a function

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𝑚𝑡 = limΔ𝑥⟶0

𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0Δ𝑥

Finally, the secant becomes the tangent and the

slope of the secant becomes the slope of the

tangent.

Spring semester 2017 Section 1: Fundamentals of differential calculus

The average rate of change of a function between 𝑃 and 𝑄 becomes the local rate of

change in 𝑃 if 𝑄 is moved towards 𝑃, i.e. if ∆𝑥 tends to 0.

Differential quotient – Definition

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The function 𝑓 is called differentiable in 𝑥0 if the

limit of the difference quotient

𝑚𝑡 = 𝑓′ 𝑥0 = limΔ𝑥⟶0

𝑓 𝑥0+Δ𝑥 −𝑓 𝑥0

Δ𝑥

exists in 𝑥0.

Our notation for this limit is 𝑓′ 𝑥0 and we call it

differential quotient,

derivative,

slope of the tangent or

local rate of change of 𝑓

in 𝑥0.

Spring semester 2017 Section 1: Fundamentals of differential calculus

𝑥0

𝑓(𝑥0)

Spring semester 2017 Section 1: Fundamentals of differential calculus

There are several notations used for the derivative.

The most widely used notation for the derivative of 𝑓 is 𝑓′. This notation was introduced

by Newton.

Since the derivative is the same as the differential quotient, the Leibniz notation is also

used quite often. Here, the derivative of 𝑓 is written as 𝑑𝑓

𝑑𝑥.

Both notations mean the same, namely the derivative of 𝑓. Therefore 𝑓′ =𝑑𝑓

𝑑𝑥.

The calculator can numerically find the derivative at a specific point 𝑥0. This is done by

using the following keys:

SHIFT 𝑑/𝑑𝑥,

enter the function,

enter the 𝑥-value where to calculate 𝑓′.

d-notation and calculator

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Differentiation rules – Review

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𝑓(𝑥) 𝑓′(𝑥) Remarks

1. 𝑐 0 𝑐 ∈ ℝ, any real constant

2. 𝑥 1

3. 𝑥𝑟 𝑟 ∙ 𝑥𝑟−1

4. 𝑐 ∙ 𝑔(𝑥) 𝑐 ∙ 𝑔′(𝑥) 𝑐 ∈ ℝ, factor rule

5. 𝑢 𝑥 + 𝑣(𝑥) 𝑢′ 𝑥 + 𝑣′(𝑥) sum rule

These rules can be expressed in a shorter way by omitting the argument x, e.g. the sum rule can be written as

𝑢 + 𝑣 ′ = 𝑢′ + 𝑣′ instead of 𝑢 𝑥 + 𝑣 𝑥 ′ = 𝑢′ 𝑥 + 𝑣′(𝑥)

Exercise – Repetition

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Find the first and second derivatives of the following functions:

a) 𝑔 𝑥 = 5

b) 𝑝 𝑥 = 𝑥2

c) 𝑓 𝑡 = 3𝑥4

d) 𝑓 𝑡 = 5𝑡3

e) ℎ 𝑢 = 3𝑢4 −1

2𝑢2 + 3

f) 𝑓 𝑥 =3𝑥2 +

1

𝑥2−

1

𝑥

Spring semester 2017 Section 1: Fundamentals of differential calculus

Product rule

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If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then 𝑓 𝑥 = 𝑢(𝑥) ∙ 𝑣(𝑥) is also

differentiable and its derivative is given by:

𝑓(𝑥)′ = 𝑢′ 𝑥 ⋅ 𝑣(𝑥) + 𝑢(𝑥) ⋅ 𝑣′(𝑥)

Spring semester 2017 Section 1: Fundamentals of differential calculus

Short notation: 𝑢 ∙ 𝑣 ′ = 𝑢′𝑣 + 𝑢𝑣′

Example: 𝑓 𝑥 = 𝑥 + 1 𝑥 − 1

𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1

𝑓′ 𝑥 = 1 ∙ 𝑥 − 1 + 𝑥 + 1 ∙ 1= 𝑥 − 1 + 𝑥 + 1 = 2𝑥

(One can reach the same result by first expanding the functional term and then differentiating the expression in its expanded form.)

Quotient rule

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If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then the function 𝑓 𝑥 =𝑢(𝑥)

𝑣(𝑥)is

also differentiable and its derivative is given by:

𝑓(𝑥)′ =𝑢′(𝑥) ⋅ 𝑣 𝑥 − 𝑢 𝑥 ⋅ 𝑣′(𝑥)

𝑣(𝑥)2

Spring semester 2017 Section 1: Fundamentals of differential calculus

Short notation :𝑢

𝑣

′=

𝑢′𝑣−𝑢𝑣′

𝑣2

Example: 𝑓 𝑥 =𝑥+1

𝑥−1

𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1

𝑓′ 𝑥 =1∙ 𝑥−1 − 𝑥+1 ∙1

𝑥−1 2 =𝑥−1−𝑥−1

𝑥−1 2 =−2

𝑥−1 2

Note theminus sign!

Chain rule

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Example:

𝑓 𝑥 = 3𝑥2 + 6𝑥 79

Outer function: 𝑢 𝑣

Inner function: 𝑣(𝑥)

Is the composition of the two functions 𝑢 ∘ 𝑣 exists and if 𝑢 is

differentiable at 𝑣(𝑥) then derivative of the composition is given by:

𝑢 ∘ 𝑣 ′ 𝑥 = 𝑢 𝑣 𝑥 ′ = 𝑢′ 𝑣 𝑥 ⋅ 𝑣′(𝑥)

𝑓′ 𝑥 = 79 ⋅ 3𝑥2 + 6𝑥 78 ⋅ 6𝑥 + 6 = ⋯

Outer derivative : 𝑢′ 𝑣 Inner derivative: 𝑣′(𝑥)

Spring semester 2017 Section 1: Fundamentals of differential calculus

Short notation : 𝑢 ∘ 𝑢 ′ = 𝑢′(𝑣) ∙ 𝑣′ In other words: outer derivative multiplied by inner derivative

Examples – Product rule, quotient rule, and chain rule

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Find the derivatives of

a) 𝑓 𝑥 = 2𝑥2 − 1 3𝑥 + 1

b) 𝑔 𝑥 =2𝑥2−1

3𝑥+1

c) ℎ 𝑥 =32𝑥2 − 1

Spring semester 2017 Section 1: Fundamentals of differential calculus

Derivatives of exponential and logarithmic functions

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For 𝑎 ∈ ℝ+\ 1 the functions 𝑎𝑥 and log𝑎𝑥 are differentiable and

their derivatives are:

a) 𝑒𝑥 ′ = 𝑒𝑥

b) 𝑎𝑥 ′ = 𝑎𝑥 ⋅ ln 𝑎

c) ln 𝑥 ′ =1

𝑥

d) log𝑎 𝑥′ =

1

𝑥⋅

1

ln 𝑎

Spring semester 2017 Section 1: Fundamentals of differential calculus

with 𝑒 ≔ 2.71828… Euler number

Examples

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Find the derivatives of

a) 𝑓 𝑥 = 𝑒−𝑥2

2

b1) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the product rule)

b2) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the quotient rule)

c) ℎ 𝑥 = log𝑎 2 𝑥

Rules of differentiation: Overview (Formulary)

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